This is kinda related -- can you say "normaal" for normally, or begin a sentence with it? Like "Normaal stel ik geen vragen"... Or is doorgaans like better for that or something?
Yes you can, your sentence is fine. Doorgaans sounds more formal and is used less I think. But I think normaal gesproken is used most when in English normally or usually is used. Furthermore you can use meestal (most of the time) for usually.
Do you not have a version of "normaalweg" as you would say in Afrikaans?
Normaalweg is not a word in Dutch, but gewoonweg is. Although it has a different meaning than normaal gesproken. It translates to simply in sentences like: Simply the best = Gewoonweg de beste.
Does this mean that the question is unusual/unexpected or does it mean more like inappropriate/indecent (like something a normal person would not ask)?